The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix of a directed graph is number of vertex-disjoint cycle covers and these are also the standard models.
Given the importance of permanent as a $\#P$ complete problem that arises in various contexts such as lattices and polyhedra are there any other non-trivial interpretations of the permanent?
I would be interested in any interpretations to number theory (I know none and if there is one it would be very interesting) or algebraic geometry (however tenuous it may be) particularly since the latter is contemplated to be useful in studying the Permanent-Determinant problem.
I would even consider this as an interpretation for special matrices. Permanent represents natural numbers in a canonical way through special matrices Is every positive integer the permanent of some 0-1 matrix?. However if there is any multiplicative and additive property that comes along with it that would be very great and so far I do not see it but it looks like it should be manageable to concoct interesting classes of matrices which have specific interpretations.
Best Answer
In quantum physics, the state of non-interacting particles that are fermions is represented by a determinant, while for bosons it is a permanent. The computational complexity of the evaluation of a permanent explains why it is possible to design a quantum computer with linear optics (LOQC) --- so with noninteracting photons, which are bosons. If the quantum computer would be evaluating determinants there would be no speedup with respect to a classical computer, so for electrons (which are fermions) one would need to resort to interactions to achieve any gain.